Minimal, rigid foliations by curves on ℂℙn
Minimal, rigid foliations by curves on ℂℙn
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ℂℙn 上的曲线呈现最小、刚性的叶状结构
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
J. Rebelo
中科院分区：
文献类型：
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作者：
F. Loray;J. Rebelo
摘要。用曲线证明了在投影空间n上每维n≥2且每阶d≥2的极小刚性奇异全纯叶的存在性。精确地说，我们构造了一个由d次齐次向量场引生的叶形（），它有一个有限奇异集，并且在整个n中所有规则叶都是稠密的。此外，满足混沌动力学的许多附加性质，并且在以下意义上是刚性的：如果通过一个足够接近恒等的同胚共轭到另一个全纯叶上，那么这些叶也通过一个投影变换共轭。最后，所有这些性质对于微小的扰动都是持久的。这是通过考虑在单位球上生成的伪群来实现的。n通过Diff（n,0）中各元素的小扰动来确定其≤n。在发生器上的开放条件下，我们证明了许多伪流的闭包（对于c0拓扑）的存在性，它们传递作用于球上。这种方法可以很容易地推导出最小性、遍历性、正熵和刚性等动力学特征。最后，在多项式向量场的横向动力学中实现了其中的一些伪群。
Abstract.We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙn for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙn. Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.¶This is done by considering pseudo-groups generated on the unit ball ?n⊆ℂn by small perturbations of elements in Diff(ℂn,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C0-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙn.